Question

NO LINKS!!!

13. Prove that the functions are inverses by showing that f(g(x))= x and g(f(x)) = x

f(x) = x^3 + 7 and g(x) = ∛(x-7)

Answer 1

See below for proof.

Explanation:

`\large\boxed{\begin{minipage}{4 cm}\underline{Exponent Rules}\\\\$\sqrt[n]{a}=a^{(1)/(n)}$\\\\$(a^b)^c=a^(bc)$\\\end{minipage}}`

Given functions:

`\begin{cases}f(x) = x^3 + 7\\g(x) = \sqrt[3]{x-7}\end{cases}`

The composite function f[g(x)] means to substitute the function g(x) in place of the x in function f(x):

`\begin{aligned}f[g(x)]&=[g(x)]^3+7\\&=(\sqrt[3]{x-7})^3+7\\&=((x-7)^(1)/(3))^3+7\\&=(x-7)^(3)/(3)+7\\&=(x-7)^1+7\\&=x-7+7\\&=x\end{aligned}`

The composite function g[f(x)] means to substitute the function f(x) in place of the x in function g(x):

`\begin{aligned}g[f(x)]&=\sqrt[3]{f(x)-7}\\&=\sqrt[3]{(x^3+7)-7}\\&=\sqrt[3]{x^3+7-7}\\&=\sqrt[3]{x^3}\\&=(x^3)^{(1)/(3)\\&=x^{(3)/(3)\\&=x^1\\&=x\end{aligned}`

Hence proving that the functions are inverses by showing that f(g(x))= x and g(f(x)) = x.